A155871 Subtraction of polynomial coefficients of MacMahon A060187 from third derivative of Pascal's triangle A155863: p(x,n)=(If[n == 0, 1, x^n + 1 + x*D[( x + 1)^(n + 1), {x, 3}]] - 2^n*(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2])/x.

1, 1, -16, -110, -16, -117, -1322, -1322, -117, -512, -9703, -22288, -9703, -512, -1843, -58977, -256363, -256363, -58977, -1843, -6048, -328588, -2477728, -4664934, -2477728, -328588, -6048, -18953, -1751300, -21692852, -69388094

Offset: 3

Author

Roger L. Bagula, Jan 29 2009

Keywords

• sign
• tabl
• uned

Comments

Row sums are:

{2, -142, -2878, -42718, -634366, -10289662, -185702398, -3715637758,

-81748930558, -1961988796414, -51011749920766}

Examples

			{1, 1},
{-16, -110, -16},
{-117, -1322, -1322, -117},
{-512, -9703, -22288, -9703, -512},
{-1843, -58977, -256363, -256363, -58977, -1843},
{-6048, -328588, -2477728, -4664934, -2477728, -328588, -6048},
{-18953, -1751300, -21692852, -69388094, -69388094, -21692852, -1751300, -18953},
{-58048, -9108221, -178273184, -906867842, -1527023168, -906867842, -178273184, -9108221, -58048},
{-175815, -46690547, -1403033205, -10836712218, -28587853494, -28587853494, -10836712218, -1403033205, -46690547, -175815},
{-529712, -237214810, -10708833968, -121383574287, -477020204064, -743288082732, -477020204064, -121383574287, -10708833968, -237214810, -529712},
{-1592125, -1198358670, -79944129566, -1295922974075, -7310749751463, -16818058154484, -16818058154484, -7310749751463, -1295922974075, -79944129566, -1198358670, -1592125}
		

Crossrefs

A060187, A155863

Programs

Mathematica

p[x_, n_] = (If[n == 0, 1, x^n + 1 + x*D[(x + 1)^(n + 1), {x, 3}]] - 2^n*(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2])/x;
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 3, 13}];
a = Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 3, 13}];
Flatten[a]

Formula

p(x,n)=(If[n == 0, 1, x^n + 1 + x*D[( x + 1)^(n + 1), {x, 3}]]

- 2^n*(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2])/x;

t(n,m)=coefficients(p(x,n))